1. AC modeling of converters containing resonant switches
State-Space Averaging: see textbook section 7.3
Averaged Circuit Modeling and Circuit Averaging: see textbook section 7.4

2. Averaged Switch Modeling
Separate switch elements from remainder of converter
Remainder of converter consists of linear circuit
The converter applies signals xT to the switch network
The switch network generates output signals xs
We have solved for how xs depends on xT

3. Block diagram of converter
Switch network as a two-port circuit:

4. The linear time-invariant network

5. The circuit averaging step
To model the low-frequency components of the converter waveforms, average the switch output waveforms (in xs(t)) over one switching period.

6. Relating the result to previously-derived PWM converter models: a buck is a buck, regardless of the switch
We can do this if we can express the average xs(t) in the form

7. PWM switch: finding Xs1 and Xs2

8. Finding µ: ZCS example
where, from previous slide,

9. Derivation of the averaged system equations of the resonant switch converter
Equations of the linear network
(previous Eq. 1):
Substitute the averaged switch
network equation:
Result:
Next: try to manipulate into same
form as PWM state-space averaged result

10. Conventional state-space equations: PWM converter with switches in position 1
In the derivation of state-space averaging for subinterval 1:
the converter equations can be written as a set of linear
differential equations in the following standard form (Eq. 7.90):
These equations must be equal:
Solve for the relevant terms:
But our Eq. 1 predicts that the circuit equations for this interval are:

11. Conventional state-space equations: PWM converter with switches in position 2
Same arguments yield the following result:
and

12. Manipulation to standard state-space form
Eliminate Xs1 and Xs2 from previous equations.
Result is:
Collect terms, and use the identity µ + µ’ = 1:
—same as PWM result, but with d  µ

13. Perturbation and Linearization
The switch conversion ratio µ is generally a fairly
complex function. Must use multivariable Taylor series,
evaluating slopes at the operating point:

14. Small signal model
Substitute and eliminate nonlinear terms, to obtain:
Same form of equations as PWM small signal model. Hence same model applies, including the canonical model of Section 7.5.
The dependence of µ on converter signals constitutes built-in feedback.

15. Salient features of small-signal transfer functions, for basic converters

16. Parameters for various resonant switch networks

17. Example 1: full-wave ZCS Small-signal ac model

18. Low-frequency model

19. Example 2: Half-wave ZCS quasi-resonant buck

20. Small-signal modeling

21. Equivalent circuit model

22. Low frequency model: set tank elements to zero

23. Predicted small-signal transfer functions Half-wave ZCS buck